### Why the risk you measure is probably not the risk you care about

##### September 26, 2022

The risk we measure is usually not the risk we care about. Why and does it matter?

Suppose we care about some investment, perhaps a loan, stock or a bond. Or even systemic risk. And the distribution of outcomes looks like:

The red line is the distribution of outcomes as they are now, that is, where we're not trying to control risk.

However, that risk may be too much so we want to shift the distribution to the blue one in the figure below:

For investors that may be achieved by several ways, perhaps derivatives, asset choices or changes in leverage. And such distribution shifts are the objective of Macro Pru.

We aspire to reduce the chance of really bad outcomes and increase the likelihood of good outcomes. Along the way shift the mean to the right.

In the technical language, thin the lower tail and fatten the upper tail.

This is really what risk management and investing amd Macro Pru is all about, shifting the distribution of outcomes to one that is preferred.

Whether for someone doing high-frequency trading, a pension fund with a multi decade horizon or the financial authorities thinking about systemic risk.

Here is the problem.

The risk lives in the middle of the distribution but what we care about is in the tails. So we end up with a reasonably good estimate of the middle, but since, by definition, there is little or no data on the tails, it is not possible to use data to estimate what happens outside of the green zone.

We measured one part of the distribution of outcomes, day-to-day risk. In doing so, we create a mathematical description of all possible outcomes — a distribution.

And when we have a mathematical distribution, we can plug in any possible combination of probabilities and outcomes you want. Now we can say, "ah, now I can measure risk over 10 years or 50 years or a decade or a millennia". The technical name for this is probability shifting.

We get a number, but is that number accurate? No. You might just as well fire up your Excel and get your risk estimate from typing in rand().

And that is why the risk we care about is usually not the risk we measure. We measure the green middle part while we really care about what's happening outside of the green zone.

And on top of that we have the problem of the distribution changing all the time. By the time we realise those tail events, the distribution will likely be quite different.

One example of The Illusion of Control.